HB 03-30-01
Resonance Tube
Resonance Tube
Lab 9
1
Lab 9
Equipment SWS, complete resonance tube (tube, piston assembly, speaker stand, piston
stand, mike with adaptors, channel), voltage sensor, 1.5 m leads (2), (room) thermometer,
flat rubber stopper
Reading Sections in your text books on waves, sound, resonance, and normal modes
1
Purpose
This lab will investigate standing sound waves in a tube subject to different boundary conditions. One end of the tube is closed off with a small speaker. The other end of the tube
can be closed off with a movable piston which can be used to change the length of the tube.
The piston can also be removed leaving one end of the tube open. Sinusoidal sound waves
are excited in the tube with the speaker. A small microphone (mike) in the tube detects the
sound waves. At the resonant frequencies of the tube the sound waves in the tube will be
enhanced.
In another experiment, a pulse of sound will be sent down the tube and the reflected pulse
examined. The nature of the reflected pulse will depend on whether the end of the tube is
open or closed. It will also be possible to make a rather direct measurement of the speed of
sound.
2
2.1
Theory
Air As A Spring
Gas is a springy material. Consider gas inside a cylinder that is closed off with a piston.
Initially, assume that the pressure on both sides of the piston is the same. If the piston is
pushed in, the entrained gas is compressed, the pressure rises, and there is a net force pushing
the piston out. If the piston is pulled out, the entrained gas is rarefied, the pressure drops,
and there is a net force pushing the piston back in. Because a gas acts like a spring and has
mass it can support oscillations and waves. In this experiment we confine our attention to
the mixture of gases we call air, which is roughly 80% N2 and 20% O2 .
2.2
Traveling Sound Waves in Air
When the cone of a loud speaker moves out, it compresses the air next to it and also
imparts an outward velocity to the air molecules. This velocity is in addition to the random
thermal velocities of the air molecules. Those air molecules nearest the speaker collide with
the adjacent air molecules and impart to those molecules their motion. In this way the
compression propagates away from the speaker, and sound is produced. Similar statements
apply to the rarefactions produced when the speaker cone moves in. If the speaker cone is
vibrated sinusoidally, a propagating sinusoidal sound wave or traveling wave moves out from
the speaker. The wave relation, λf = v, is satisfied by this wave. Here, λ is the wavelength,
f is the frequency, and v is the wave speed. As the motion of the air molecules is along
the direction in which the wave propagates, the waves are called longitudinal waves. This is
in contrast to the waves on string which are called transverse waves as the elements of the
string move transverse to the direction in which the wave travels.
HB 03-30-01
Resonance Tube
Lab 9
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In traveling sound waves the displacement of the air particles satisfies the wave equation.
It is found that the velocity of the waves v is given by
s
v=
γp
ρ
(1)
where p is the air pressure and ρ is the air mass density. The quantity γ is the ratio of the
specific heats at constant pressure and constant volume, CP /CV . The quantity γ appears
because the compressions and rarefactions that occur in sound waves are so fast that the
process is adiabatic (there is no heat flow). For air, which is composed mostly of diatomic
molecules, γ = 1.4.
With the aid of the ideal gas law, Eq.(1) can be written as
s
γRT
,
(2)
M
where R is the molar gas constant, T is the absolute temperature, and M is the molar mass.
For a given gas the speed is only proportional to the square root of the temperature. Some
insight into this fact is given by noting that the root mean square thermal speed (vrms ) of
the molecules in a gas is
v=
s
3RT
.
(3)
M
The speed of sound in a gas is quite close to the thermal speed of the molecules in the gas.
The disturbance of the wave is propagated by the molecules colliding with each other, and
the velocity of propagation is essentially the thermal speeds of the molecules.
A convenient expression for the speed of sound v in air at room temperatures is given by
vrms =
v = 331.5 + .606T m/s,
(4)
where T is the temperature in centigrade.
2.3
Traveling Sound Waves in a Tube
Sound waves can travel down a tube of constant cross section very much as they do in free
space. It is assumed here that the walls of the tube are rigid so that they do not flex under
the pressure variations of the wave, and that the walls of the tube are smooth so that there
is not much attenuation of the wave. The speed of the waves in the tube is essentially the
same as in free space.
2.4
Standing Sound Waves in a Finite Tube
Traveling sound waves in a finite tube are reflected at the ends of the tube. If the tube length
and boundary conditions at the two ends of the tube are appropriate, resonance can occur at
certain frequencies which are called resonant frequencies. (These are also the normal modes
of the air entrained in the tube.) Resonance occurs when the reflected waves at the two
ends of the tube reinforce each other. A small excitation builds up a large standing wave
resonance.
The “pressure” of the air in the wave is understood to be the change in the pressure
from the average value. The “displacement” of the air in the wave is understood to be the
HB 03-30-01
Resonance Tube
Lab 9
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displacement of the air from its equilibrium position. At resonance, both the pressure and
displacement vary sinusoidally in space and time. Points in the tube where the pressure variations are maximum are called pressure antinodes, and points in the tube where the pressure
variations are zero are called pressure nodes. Points where the displacement variations are
maximum are called displacement antinodes, and points where the displacement variations
are zero are called displacement nodes. In standing sound waves pressure nodes occur at
displacement antinodes and pressure antinodes occur at displacement nodes.
The open end of finite tube is a pressure node because of the reservoir of normal air
pressure outside the tube. (Actually the pressure node occurs a bit outside the end of the
tube.) The same point must be a displacement antinode. The end of a closed tube must be a
displacement node and a pressure antinode. From these boundary conditions the resonance
frequencies of a tube open at both ends, closed at both ends, and open at one end and closed
at the other end, can be calculated. A tube with both ends open is called an open tube.
A tube with one end open and one end closed is called a closed tube. Fig. 1 shows the air
displacement for the first 4 resonances of an open tube and a closed tube. The resonance
wavelengths are determined by fitting standing waves into the tube so that the boundary
conditions at both ends are satisfied. The lowest resonant frequency is called the fundamental
or 1st harmonic. The nth harmonic is n times the fundamental. Not all harmonics need be
present.
3
Apparatus
The resonance tube is shown assembled in Fig. 2, and the parts are shown in Fig. 3. The
tube has a built in metric scale. The speaker-microphone stand is affixed at one end of the
tube and secured by shock cord. This stand has a vertical plate that holds the speaker. A
hole in the plate accommodates the small microphone (mike). Note that the mike is sensitive
to pressure variations. A thumbscrew holds the microphone in place. PLEASE TIGHTEN
THIS THUMBSCREW ONLY SLIGHTLY. Always make sure that the plate holding the
speaker and mike presses firmly against the tube.
For the pulse experiments it is necessary to move the mike into the tube to the 10 cm
mark. To do this, loosen the thumbscrew holding the mike. Move the speaker stand out
a few cm from the resonance tube and then push the mike by its cord into the resonance
tube. Then place the speaker stand back firmly against the resonance tube. This procedure
is necessary because the mike just catches the inner diameter of the resonance tube.
The other end of the resonance tube is held by the piston stand, also secured by shock
cord. The piston fits inside the tube and is attached to a rod that goes through a vertical
plate in the stand. The position of the piston is determined by the scale on the resonance
tube.
For one experiment, the piston and its stand are removed from the end of the resonance
tube. There is a piece of channel which can be used to support the end of the resonance
tube for this experiment. This piece of channel can also be used to support the end of the
piston rod when the piston is in the tube.
The lead from the mike goes to a small battery powered preamplifier which has an off-on
switch. At the start of the experiment, turn the switch on (red showing next to switch).
To conserve the battery, turn the switch off when you are through. The output of the
preamplifier goes to a phone plug. There are adaptors that receive the bannana plugs of a
voltage sensor.
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Resonance Tube
Lab 9
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The speaker is driven by the power amplifier sitting on top of the interface. Check
that the output of the power amplifier is connected to the speaker by two leads. DO NOT
OVERDRIVE THE SPEAKER. Limit voltages to the speaker to 2 V. You should hear sound,
but it should not be very loud. Connect a voltage sensor to channel A of the interface and
to the output of the preamplifier.
Program the interface as follows.
• Drag the analog plug icon to channel C and choose power amplifier. The voltage out
button is automatically depressed.
• Drag the analog plug icon to channel A and choose voltage sensor.
• Double click the voltage sensor icon to open the voltage sensor setup window. Change
the sensitivity from the default of “low” to medium. (SWS has an analog to digital
converter (ADC) of 212 = 4096. At a sensitivity of low, voltages from −10 to +10 volts
are digitized by dividing this voltage interval up into 4096 steps. Each step is about 5
mV. At medium sensitivity the voltage interval is −1 to +1, and each step is about 0.5
mV. At high sensitivity, the voltage interval is −0.1 to +0.1 and each step is 0.05 mV.)
• Drag the scope icon to the voltage output terminals. The signal generator voltage is
now applied to the top channel (green) of the scope. This is also the channel that the
scope triggers on. (It is preferable to trigger on the larger voltage driving the speaker
than on the much smaller preamplifier voltage.)
• Click the input menu button for the middle channel (red) of the scope and choose
channel A, or mike output.
4
Using the SWS Scope
The SWS oscilloscope can be used to observe the waveform of the sound waves. The x axis of
the scope is time. The y axis is the voltage input. Practice using the following components
of the SWS scope. See figure 6 for the scope diagram.
A. The drag down menu is used to select the input for that color (green, red, or blue) scope
trace. Up to three inputs can be selected at a time, and will be shown simultaneously
on the scope.
B. These two buttons change the y-axis scale, either stretching the scope trace or shrinking
the trace in the vertical direction.
C. Moves the scope trace up or down on the screen.
D. Changes the x-axis scale, either stretching or compressing the trace in the horizontal
direction.
E. This button selects the cross hairs function. The cross hairs function will allow you to
determine the coordinates of any point on the scope. The time is displayed at the
bottom of the screen. The y-axis value is displayed next to button A.
F and G. The trigger is used to make the scope trace appear static on the screen. If you
notice that the trace is “moving” across the screen, turn on the trigger button. Move
the green triangle G up or down to freeze the trace. The green triangle changes the
trigger level, the value at which the sweep will start on the signal.
HB 03-30-01
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Resonance Tube
Lab 9
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Experiment: Measuring Wavelength
Set the signal generator for a 500 Hz sine wave with an amplitude of 1.0 V. Click AUTO
and MON and adjust the scope so that both traces are observable. With the piston in
the tube, determine the positions of the piston that result in maximum signal from the
mike. These are the resonant frequencies of the tube. Start with the resonance for the
shortest tube (piston nearest the speaker). Make a number of measurements so as to
build up statistics and to determine how well you can make the measurement. Then
move the piston out, determining which piston position results in the next resonance.
For this frequency, you will be able to adjust for only two resonances. Successive
piston positions for resonance are a half wavelength apart. Determine the wavelength
λ from your data, averaging appropriately. Compare your measured wavelength to the
theoretical value, given by λf = v.
In units of wavelength, how close is the speaker to a displacement minimum or maximum? Can you make a comparison to the driven string experiment?
Repeat the above measurements for frequencies of 1000 Hz and 1500 Hz, and make the
same analysis. These are shorter wavelengths and more resonances will be observable.
Do the wavelengths change with frequency in the way you expect?
6
Pulsed Experiments
In these experiments, short pulses will be sent down the tube from the speaker. This is
achieved by driving the speaker with a relatively low frequency square wave of 10 Hz.
The speaker cone move very quickly one way and then stops. This sends a short
compression or rarefaction pulse down the tube. This pulse is reflected from the end of
the tube and propagates back toward the speaker. It bounces off the speaker end of the
tube, and continues to go up and down the tube until it is damped. The period of the
square wave is chosen long enough so that the pulse is completely dissipated before the
speaker cone moves the other way in response to the square wave and sends the next
pulse down the tube. The pulses alternate between compressions and rarefaction.
6.1
Speed of Sound
In this section, think carefully about what you are seeing on the scope. If you completely
understand what you see, you understand what an oscilloscope is doing. Set the signal
generator for a 10 Hz symmetric square wave with an amplitude of 2 V. Loosen the
thumbscrew holding the mike, back off the speaker stand, and slide the mike into the
tube to the 10 cm mark on the tube scale. Push the speaker stand back against the
tube. Move the piston out to 80 cm, supporting the end of the piston rod with the
channel. Click MON and adjust the scope so that the pattern looks something like
Fig. 4. Explain the scope trace. Now increase the scope sweep speed so that the trace
looks like Fig. 5. You may need to switch the speaker leads to invert the trace. The
first pulse after the change in the voltage of the square wave is the outgoing pulse
passing over the mike. The second pulse is the reflected pulse passing over the mike. Is
the reflected pulse inverted? By using the stated horizontal sweep speed of the scope,
determine the speed of sound. You will need to choose a point in the pulse which can
HB 03-30-01
Resonance Tube
Lab 9
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be identified with minimum error. In your report, state which point in the pulse you
chose.
With the scope running, slowly move the piston toward the speaker. What do you see
on the scope? Explain.
6.2
Boundary Conditions
Very generally, when a wave or a pulse hits the boundary of the medium in which it has
been traveling, it is partially reflected and partially transmitted. An example would be
light impinging on glass. For some boundary conditions there is no transmitted wave.
An example would be light that undergoes total internal reflection. For the resonance
tube, this will be the case for the boundary conditions which will examined. The nature
of the reflected pulse depends on the boundary conditions. The pulse that is reflected
must have a form so that the boundary conditions are satisfied. If the end of the tube
where the pulse is reflected is open, the pressure in the reflected pulse must be inverted
so as to cancel the pressure of the incoming pulse. The end of the tube is a pressure
node. If the end of the tube where the pulse is reflected is closed, the pressure in
the reflected pulse is not inverted. The end of the tube is a displacement node and a
pressure antinode.
Remove the piston and piston stand from the tube and support the end of the tube with
the channel. Observe the nature of the reflected pulse. Now close the end of the tube
with the rubber stopper provided and note the change in the nature of the reflected
pulse. Discuss what you see.
With the stopper in place on the end of the tube, try “cracking” open the end of the
tube a bit by bending the stopper. How much of a crack is necessary to change the
reflected pulse?
7
Questions
1. Let the length of the tubes in Fig. 1 be L. What are the wavelengths at the fundamental frequencies of the two types of tubes?
2. What is the wavelength at the fundamental frequency of a tube of length L closed
at both ends?
3. Show that Eq.(2) follows from Eq.(1).
8
Finishing Up
Please leave the bench as you found it. TURN OFF THE MIKE PRE-AMPLIFIER
SO AS TO CONSERVE THE BATTERY. Thank you.
HB 03-30-01
Resonance Tube
Lab 9
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B
C
G
D
E
F
Figure 6